Trace Representation of Primitive -LFSR Sequences and Its Application

-LFSR is a kind of word-oriented LFSR with high efficiency and good cryptographic properties, especially suitable for modern processors. It can be used in stream cipher for fast software implementation. But in practicality, primitive -LFSR sequences are of the most importance. Firstly, by the iterative relationship of the primitive -LFSR sequences and the trace function in finite fields, the explicit expression of primitive -LFSR sequences is presented. Therefore the trace representation is gotten; then a sufficient and necessary condition is obtained due to the trace representation. It can be used to check whether a -LFSR sequence is primitive or not. They provide the new tool for further research of primitive -LFSR sequences.